The Magpies responded to their disappointing defeat at Brentford two weeks ago with a wonderful display and a deserved three points at St. James' Park, with Barnes the star man in the second half.
He could - and perhaps should - have given his side the lead inside the first minute. Bravely pressing high up the field, United won possession from a City goal kick and Joelinton fed Barnes but the in-form wideman saw his somewhat tame effort saved by Gianluigi Donnarumma.
The Magpies' bold approach then almost led to their undoing two minutes later, when possession was conceded in midfield and a simple pass from Jérémy Doku slided open an exposed home backline. The Premier League's top scorer, Erling Haaland, raced onto the ball but sliced his shot horribly off target.
Nick Pope comfortably held a Rayan Cherki strike after the France international cut inside from the left, then in the 13th minute Donnarumma pulled off an outstanding save to deny Nick Woltemade.
Jacob Murphy's delivery from the right was sublime and the big German met it perfectly with his head, but the visiting goalkeeper parried brilliantly at the Leazes End.
Referee Sam Barrott waved away a penalty appeal for both sides - Lewis Hall went down under a challenge from Rúben Dias then at the other end Phil Foden shot wide under pressure from Fabian Schär - before Donnarumma saved City once more.
In the 26th minute, Barnes played in Woltemade but the Newcastle centre-forward's powerful low strike from inside the box was blocked by the Italy number one.
Five minutes later, another gilt-edged chance came United's way, as Murphy picked out Barnes at the far post, but the former Leicester City man, who scored his 50th Premier League goal at Brentford a fortnight earlier, was unable to make it 51 as he fired wide.
As a thrilling first half continued, Cherki was off target for the visitors from distance after some pinball in the Newcastle box, then Pope made a great reaction save from Haaland after Nico O'Reilly had cut the ball back for the Norwegian and Foden shot wide after exchanging passes with Cherki.
Right at the beginning of the second half, Barnes forced a corner and Sandro Tonali's flag kick flashed across goal.
Bernardo Silva was off target for the visitors but while Pep Guardiola's men probed, it was Newcastle who looked more likely to break the deadlock.
Woltemade was thwarted again by Donnarumma, who kept out the striker's effort after an incisive first-time centre from Murphy.
But the goalkeeper was finally beaten in the 64th minute when Bruno Guimarães found Barnes, who gave the ball back to the Brazilian. He then poked the ball back into the path of Barnes, who drilled low and hard into the Gallowgate net.
City responded quickly, as United failed to clear a corner allowing Dias to rifle home via a deflection.
But when Woltemade headed across goal and Guimarães hit the bar, Barnes showed composure and quality to take a touch and drill beyond Donnarumma. There was a lengthy VAR check for a potential offside, but referee Barrott eventually awarded the goal.
Eight minutes of additional time made for a nervy finale, but United saw out the game to seal three points and a century of wins as Newcastle boss for Howe.
